Báo cáo toán học: "Oriented Matroids Today" pptx

Oriented matroids are both important and interesting objects of study in Combinatorial Geometry, andindispensable tools of increasing importance and applicability for many other parts of

G¨ unter M Ziegler∗

Department of Mathematics, MA 7-1 Technische Universit¨ at Berlin Strasse des 17 Juni 136

10623 Berlin, Germany ziegler@math.tu-berlin.de http://www.math.tu-berlin.de/ ∼ziegler

Submitted: October 6, 1995; Accepted: March 26, 1996; Version 3 of September 10, 1998.

Mathematics Subject Classification: 52-00 (52B05, 52B30, 52B35, 52B40)

Abstract This dynamic survey offers an “entry point” for current research in oriented matroids For this, it provides updates on the 1993 monograph “Oriented Matroids” by Bj¨ orner, Las Vergnas, Sturmfels, White & Ziegler [85], in three parts:

1 a sketch of a few “Frontiers of Research” in oriented matroid theory,

2 an update of corrections, comments and progress as compared to [85], and

3 an extensive, complete and up-to-date bibliography of oriented matroids, comprising and extending the bibliography of [85].

Oriented matroids are both important and interesting objects of study in Combinatorial Geometry, andindispensable tools of increasing importance and applicability for many other parts of Mathematics.The main parts of the theory and some applications were, in 1993, compiled in the quite comprehensivemonograph by Bj¨orner, Las Vergnas, Sturmfels, White & Ziegler [85] For other (shorter) introductionsand surveys, see Bachem & Kern [35], Bokowski & Sturmfels [146], Bokowski [116], Goodman & Pollack[333], Ziegler [712, Chapters 6 and 7], and, most recently, Richter-Gebert & Ziegler [565]

This dynamic survey provides three parts:

1 a sketch of a few “Frontiers of Research” in oriented matroid theory,

2 an update of corrections, comments and progress as compared to [85], and

3 an extensive, complete and up-to-date bibliography of oriented matroids, comprising and ing the bibliography of [85]

extend-∗Supported by a DFG Gerhard-Hess-Forschungsf¨orderungspreis

1

2 What is an Oriented Matroid?

Let V = (v1, v2, , vn) be a finite, spanning, sequence of vectors in R

r

, that is, a finite vectorconfiguration With this vector configuration, one can associate the following sets of data, each ofthem encoding the combinatorial structure of V

• The chirotope of V is the map

χV :{1, 2, , n}r −→ {+, −, 0}

(i1, i2, , ir) 7−→ sign(det(vi1, vi2, , vir))that records for each r-tuple of the vectors whether it forms a positively oriented basis of R

r

, abasis with negative orientation, or not a basis

• The set of covectors of V is

that is, the set of all partitions of V (into three parts) induced by hyperplanes through the origin

• The collection of cocircuits of V is the set

,

of all partitions by “special” hyperplanes that are spanned by vectors of the configuration V

• The set of vectors of V is

V(V ) :=  sign(λ1), , sign(λn)

∈ {+, −, 0}n

: λ1v1+ + λnvn= 0 is a lineardependence between vectors in V

is given, one can from this uniquely reconstruct all the others

Furthermore, one has axiom systems (see [85, Chap 3]) for chirotopes, covectors, cocircuits, vectorsand circuits that are easily seen to be satisfied by the corresponding collections above Thus there arecombinatorial structures, called oriented matroids, that can equivalently be given by any of thesefive different sets of data, and defined/characterized in terms of any of the five corresponding axiomsystems (The proofs for the equivalences between these data sets resp axiom systems are not simple.)

Vector configurations as discussed above give rise to oriented matroids of rank r on n elements (or: on

a ground set of size n) Usually the ground set is identified with E ={1, 2, , n}

Equivalent to vector configurations, one has the model of (real, linear, essential, oriented) hyperplanearrangements: finite collectionsA := (H1, H2, , Hn) of hyperplanes (linear subspaces of codimensionone) inR

r

, with the extra requirement that H1∩ .∩Hn ={0}, and with a choice of a positive halfspace

H+ for each of the hyperplanes In fact, every vector configuration gives rise to such an arrangement

in Oriented Matroid Theory:

• The Topological Representation Theorem (see [85, Chap 5]) shows that while real vector rations can equivalently be represented by oriented arrangements of hyperplanes, general orientedmatroids can be represented by oriented arrangements of pseudo-hyperplanes

configu-• There is no finite set of axioms that would characterize the oriented matroids that are sentable by vector configurations In fact, even for r = 3 there are oriented matroids on nelements that are minimally non-realizable for arbitrarily large n

repre-• The realization problem is a difficult algorithmic task: for a given oriented matroid, to decidewhether it is realizable, and possibly find a realization This statement is a by-product of theconstructions for the Universality Theorem for oriented matroids, see below

Currently there is substantial research done on a variety of aspects and questions; among them areseveral deep problems of oriented matroid theory that were thought to be both hard and fundamental,and are now gradually turning out to be just that

Here I give short sketches and a few pointers to the (recent) literature, for just a few selected topics.(By construction, the selection is very much biased I plan to expand and update regularly Your helpand comments are essential for that.)

3.1 Realization spaces.

Mn¨ev’s Universality Theorem of 1988 [503] states that every primary semialgebraic set defined overZ

is “stably equivalent” to the realization space of some oriented matroid of rank 3 In other words, thesemialgebraic sets of the form

R(X) := {Y ∈R

3 ×n: sign(det(X

i,j,k)) = sign(det(Yi,j,k)) for all 1≤ i < j < k ≤ n},for real matrices X ∈R

3 ×n, can be arbitrarily complicated, both in their topological and their

arith-metic properties Mn¨ev’s even stronger Universal Partition Theorem [504] announced in 1991 says thatessentially every semialgebraic family appears in the stratification given by the determinant function

on the (3× 3)-minors of (3 × n)-matrices

These results are fundamental and far-reaching For example, via oriented matroid (Gale) duality theyimply universality theorems for d-polytopes with d+4 vertices

For some time, no complete proofs were available This has only recently changed with the completeproof of the Universal Partition Theorem by G¨unzel [301] and by Richter-Gebert [557] Richter-Gebert[556, Sect 2.5] has also — finally! — provided a suitable notion of “stable equivalence” of semialgebraicsets that is weak enough to make the Universality Theorems true, and strong enough to imply bothhomotopy equivalence and arithmetic equivalence (i.e., it preserves the existence of K-rational points

in the semialgebraic set for every subfield K ofR)

Further, surprising recent progress is now available with Richter-Gebert’s [556, 564] Universality orem (and Universal Partition Theorem) for 4-dimensional polytopes, and related to this his Non-Steinitz theorem for 3-spheres (See [302] for a second proof.)

The-For still another, very recent, interesting universality result, concerning the configuration spaces ofplanar polygons, see Kapovich & Millson [411] Kapovich & Millson trace the history of their resultback to a universality theorem by Kempe [419] from 1875!

Here are two major challenges that remain in this area:

• To construct and understand the smallest oriented matroids with non-trivial realization spaces.The smallest known examples are Suvorov’s [640] oriented matroid of rank 3 on 14 points with

a disconnected realization space (see also [85, p 365]), and Richter-Gebert’s [558] new example

Ω+14with the same parameters, which additionally has rational realizations, and a non-realizablesymmetry

• To provide Universality Theorems for simplicial 4-dimensional polytopes (The Kleinschmidt polytope [122] is still the only simplicial example known with a non-trivial realiza-tion space; see also Bokowski & Guedes de Oliveira [124].)

Bokowski-Ewald-3.2 Extension spaces, combinatorial Grassmannians, and matroid bundles

Thanks to Laura Anderson for help on this section!

The consideration of spaces of oriented matroids brings several very different lines of thinking into

a common topological framework Given a set S of oriented matroids, we obtain a partial order

on S by weak maps, and from this we obtain a topological space by taking the order complex (thesimplicial complex given by chains in the partial order; see Bj¨orner [82]) This simplicial complex can beviewed as a combinatorial analog to a vector bundle Just as a vector bundle represents a continuousparametrization of a set of vector spaces, this topological space can be viewed as a “continuous”parametrization of elements of S Such spaces have arisen in several contexts:

• If S is the set of non-trivial single-element extensions of a fixed oriented matroid M, the resultingspace is the extension space E(M) of M

• If S is the set of all rank r oriented matroids on a fixed set of n elements, this space is theMacPhersonian MacP(r, n)

• If S is the set of all rank r strong map images of a fixed oriented matroid M, this space is thecombinatorial GrassmannianG(r, M) (In fact, this example essentially encompasses the previoustwo: The extension spaceE(M) of a oriented matroid M is a double cover of G(r(M) − 1, M),while if M is the unique rank n oriented matroid on a fixed set of n elements, then G(r, M) =MacP(r, n).)

Extension spaces are closely related to zonotopal tilings (via the Bohne-Dress Theorem) and to orientedmatroid programs: see Sturmfels & Ziegler [639] The MacPhersonian and combinatorial Grassmannianarise in MacPherson’s theory of combinatorial differential manifolds and matroid bundles [476] [16])

in which oriented matroids serve as combinatorial analogs to real vector spaces

Among the basic conjectures in the field are:

• For a rank n oriented matroid Mn, the topology of G(k, Mn) should be similar to that of thereal Grassmannian G(k,R

• The extension space E(Mn

) should have the homotopy type of an (n− 1)-sphere if Mn

is izable (This is essentially a special case of the above.)

real-There are substantial grounds for pessimism on both conjectures For instance, there are examples

of non-realizable Mn such that G(n − 1, Mn) and E(Mn) are not even connected (Mn¨ev & Gebert [505]) In addition, Mn¨ev’s Universality Theorem implies that for realizable Mn the inverseimages under c of points inG(r, Mn) can have arbitrarily complicated topology However, substantialprogress has been made on the topology of E(M) for realizable M [639] [505] and on the topology

Richter-of G(k, M) under various conditions: for small values of k (Babson [29]), for the first few homotopygroups of the MacPhersonian (Anderson [15]), and for mod 2 cohomology (Anderson & Davis [18]).Three related survey articles are Mn¨ev & Ziegler [506], Anderson [16], and Reiner [546]

The analogy between oriented matroids and real vector bundles leads to an intriguing and usefulinterplay between topology and combinatorics On the one hand, appropriate combinatorial adapta-tions of classical topological methods for real vector bundles prove that for realizable Mn the map

c : G(k,R

n

)→ G(k, Mn) induces split surjections in mod 2 cohomology [18]) On the other hand, binatorial methods can be applied to topology as well Any real vector bundle over a triangulated basespace can be “combinatorialized” into a matroid bundle [476] [18], giving a combinatorial approach tothe study of bundles The most notable success in this direction has been Gel’fand & MacPherson’s[308] combinatorial formula for the rational Pontrjagin classes of a differential manifold

com-The topological problems discussed in this section have close connections to classical problems oforiented matroid theory, such as the following: Las Vergnas’ conjectures that every oriented matroidhas at least one mutation (simplicial tope) and that the set of uniform oriented matroids of rank r on agiven finite set is connected under performing mutations In fact, if these conjectures are false, then the

“top level” of the MacPhersonian, given by all oriented matroids without circuits of size smaller than

r and at most one circuit of size r, cannot be connected As for the Las Vergnas conjecture, Bokowski[114] and Richter-Gebert [552] have the strongest positive resp negative partial results; more work isneccessary

Further work also remains in the understanding of weak and strong maps — currently the only prehensive source is [85, Section 7.7] One still has to derive structural information from the failure

com-of Las Vergnas’ strong map factorization conjecture (disproved by Richter-Gebert in [552]) and derivecriteria for situations where factorization is possible

3.3 Affine and infinite oriented matroids.

The Bohne-Dress Theorem, announced by Andreas Dress at the 1989 “Combinatorics and Geometry”Conference in Stockholm, provides a bijection between the zonotopal tilings of a fixed d-dimensionalzonotope Z and the single-element liftings of the realizable oriented matroid associated with Z Thistheorem turned out to be, at the same time,

• fundamental (see e g the connection to extension spaces of oriented matroids [639]),

• “intuitively obvious” (just draw pictures!), and

• surprisingly hard to prove; see Bohne [105] and Richter-Gebert & Ziegler [563]

Just recently, however, a new and substantially different proof of the Bohne-Dress theorem has becomeavailable, by Huber, Rambau & Santos [387] In particular, there are bijections



←→

extensions of the dualoriented matroid M∗(A)

On the other hand, there is a definite need for a better understanding of zonotopal tilings of theentire plane (or ofR

is just as hard as the “Existential Theory of the Reals,” the problem of solving general systems ofalgebraic equations and inequalities over the reals While it is not known whether the problem overQ

is at all algorithmically solvable (see Sturmfels [630]), there are algorithms available that (at leasttheoretically) solve the problem over the reals For the general problem Basu, Pollack & Roy [50]currently have the best result:

Let P = {P1, , Ps} be a set of polynomials in k < s variables each of degree at most d andeach with coefficients in a subfield K⊆R

There is an algorithm which finds a solution in each connected component of the solution set, foreach sign condition on P1, , Ps, in at most O(s)k

s dO(k)= (s/k)ks dO(k) arithmetic operations

in K

However, until now this is mostly of theoretical value What can be done for specific, explicit, smallexamples? Given an oriented matroid of rank 3, it seems that

• the most efficient algorithm (in practice) currently available to find a realization (if one exists)

is the iterative “rubber band” algorithm described in Pock [532]

• the most efficient algorithm (in practice) currently available to show that it is not realizable (if itisn’t) is the “binomial final polynomials” algorithm of Bokowski & Richter-Gebert [130] whichuses solutions of an auxiliary linear program to construct final polynomials (An explicit example

of a non-realizable oriented matroid Ω−14without a biquadratic final polynomial was just recentlyconstructed by Richter-Gebert [558].)

Neither of these two parts is guaranteed to work: but still the combination of both parts was goodenough for a (still unpublished) complete classification of all 312,356 (unlabeled reorientation classes

of) uniform oriented matroids of rank 3 on 10 points into realizable and non-realizable ones (Bokowski,Laffaille & Richter-Gebert [128]).

A very closely related topic is that of Automatic Theorem Proving in (plane) geometry In fact,the question of validity of a certain incidence theorem can be viewed as the realizability problem for(oriented or unoriented) matroids of the configuration Richter-Gebert’s Thesis [550] and Wu’s book[692] here present two recent (distinct) views of the topic, both with many of its ramifications.Here we are far from having reached the full scope of current possibilites For an (impressive) demon-stration I refer to the spectacular new Interactive Geometry Software system Cinderella by UlrichKortenkamp and J¨urgen Richter-Gebert [561], whose prover includes the idea of “binomial proofs”[207] as well as new randomized methods An amazing piece of work!

In this section, I collect some notes, additions, corrections and updates to the 1993 book by Bj¨orner,Las Vergnas, Sturmfels, White & Ziegler [85] The list is far from complete (even in view of the pointsthat I know about), and with your help I plan to expand it in the future

Page 150, Section 3.9 “Historical Sketch”

Jaritz [401, 402] gives a new axiomatic of oriented matroids in terms of “order functions” whose axiomsand concepts she traces back to Sperner [611] (1949!), Karzel [416] etc At the same time, Kalhoff[407] reduces embedding questions about pseudoline arrangements, as solved by Goodman, Pollack,Wenger & Zamfirescu [338, 341], back to 1967 results of Prieß-Crampe [537]

All this gets us closer to confirming the suspicion that probably Hilbert knew about oriented troids

ma-Page 220, Exercise 4.28∗

Part (a) of this was already proved by Zaslavsky [694, Sect 9] However, part (b) remains open andshould be an interesting challenge

Page 227, Definition 5.1.3

For condition (A2), if SA∩ Se= S−1=∅ is the empty sphere in a zero sphere SA∼= S0, then the sides

of this empty sphere are the two points of SA

log2sn < 0.6988 n2

Page 275:

Richter-Gebert [560] has proved (in 1996, and written up in 1998) that orientability is NP-complete[560] (It’s a beautiful paper!)

Page 279, Exercises 6.21(a)( ∗)

The answer is “yes”: this problem was solved in 1997, with an explicit construction, by Forge &Ram´ırez Alfons´ın [281]

Page 334, Exercises 7.15(b)(∗) and 7.17

An explicit example of an oriented matroid that has a simple adjoint, but not a double adjoint wasconstructed by Hochst¨attler & Kromberg [383, 436]

Also, they observed [382, 436] that some assertions in Exercise 7.17 are not correct: J¨urgen Gebert’s [550, p 117] 8-point torus is realizable over an ordered skew field, but not overR Thereforethe oriented matroid given by such a skew realization has an infinite sequence of adjoints, but it is notrealizable inR

Richter-4

Page 337, Exercises 7.44*

No one seems to remember the example: so consider this to be an open problem (The non-existence

of such an example is also discussed, as a Conjecture of Brylawski, in McNulty [493].)

Page 385, McMullen’s problem on projective transformations

Forge & Schuchert [282] have found a configuration of 10 points in general position in affine 4-spacethat no projective transformation can put into convex position This solves McMullen’s problem for

It is not true that the sphereS = M9

963is neighborly: the edges 13 and 24 are missing (in the labelingused in [85]) Thus Shemer’s Theorem 9.4.13 cannot be applied here A proof that the sphere admits atmost one matroid polytope, AB(9), was given by Bokowski [109] in 1978 (see also Altshuler, Bokowski &Steinberg [12] and Antonin [20]) It is described in detail in Bokowski & Schuchert [137] (The orientedmatroid RS(8) discussed in [85, Sect 1.5] arises as a contraction of the oriented matroid AB(9).)

Page 413, Exercise 9.12( ∗).

Bokowski & Schuchert [137] showed that the smallest example (both in terms of its rank r = 5 and interms of its number of vertices n = 9), is given by Altshuler’s sphere M9

Page 426, Proof of Corollary 10.1.10.

“Orthogonality of circuits and cocircuits”

5 The Bibliography.

The purpose of the following is to keep the bibliography of the book [85] up-to-date electronically.For this, the following contains all the references of this book (including those which are not directlyconcerned with oriented matroids) Into this I have inserted all the corrections, missing references,additions and updates that I am currently aware of Any corrections, new papers concerned withoriented matroids, and other updates that you tell me about will be entered asap I am eager to hearabout your corrections, updates and comments!

Related bibliographies on the web are:

• Bibliography of signed and gain graphs, by Thomas Zaslavsky, published as a dynamic surveyDS8 in the the electronic journal of combinatorics 3 (1996), DS#4; published July 20, 1998,http://www.combinatorics.org/Surveys/index.html

• Bibliography of matroids, by Sandra Kingan, at

http://members.aol.com/matroids/biblio.htm

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